Answer
$$\int\frac{e^tdt}{e^{2t}+3e^t+2}=\ln|e^t+1|-\ln|e^t+2|+C$$
Work Step by Step
$$I=\int\frac{e^tdt}{e^{2t}+3e^t+2}$$
Set $u=e^t$, which means $$du=e^tdt$$
Therefore, $$I=\int\frac{du}{u^2+3u+2}$$ $$I=\int\frac{du}{(u+1)(u+2)}$$ $$I=\int\frac{(u+2)-(u+1)}{(u+1)(u+2)}du$$ $$I=\int\frac{du}{u+1}-\int\frac{du}{u+2}$$ $$I=\ln|u+1|-\ln|u+2|+C$$ $$I=\ln|e^t+1|-\ln|e^t+2|+C$$
